Dictionary

fractional ideal en · NOUN

Meanings

1. (qualifier:ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.
   • Products of fractional ideals are again fractional ideals, since if A#92;alpha#92;subseteqR and B#92;beta#92;subseteqR, then (AB)(#92;alpha#92;beta)#92;subseteqR. 1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476:
   • 2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106, A fractional ideal of F is a finitely generated 𝒪_F-submodule of F. Fractional ideals form a group together with the ideal multiplication. The neutral element is 𝒪_F and the inverse of a fractional ideal a⊂F is a⁻¹=x∈F;xa⊂𝒪_F. […] Two fractional ideals a,b are called equivalent, if there is a r∈F such that a=rb.
   • Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group. 2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10:

Forms

Synonyms

rI ⊆ R (invertible ideal) · R-submodule of Frac(R) such that for some nonzero r∈R (invertible ideal)